E-VB satellite of the 16 th ICQC June 2018, Marseille, France. Content

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1 E-VB satellite of the 16 th ICQC June 2018, Marseille, France VB with XMVB Content STARTING WITH XMVB 2 VB COMPUTER EXERCISES 5 Exercise 1. The HF molecule. 5 Exercise 2. Ozone and sulfur dioxide 7 Exercise 3. Benzene and Anthracene 8 Exercise 4. Identity hydrogen abstraction reaction 10 Exercise 5. The H 3Si- H vs. H 3Si- Cl Bonds 11 Exercise 6. Cl- CH 3- Cl vs. Cl- SiH 3- Cl 12 Exercise 7. Allyl Radical 12 Exercise 8. CycloButadiene 13 Exercise 9. Resonance energy of formamide 13 Exercise 10. Diels- Alder reaction of 1, 3- butadiene and ethylene 14 VB PAPER EXERCISES 15 Exercise 1. The lone pairs of H 2 O 15 Exercise 2. Allyl radical 15 Exercise 3. Rumer basis set of VB structures 16 Exercise 4 : Radical character of Ozone 17 ANSWERS TO THE VB PAPER EXERCISES 18 Exercise 1. The lone pairs of H 2 O 18 Exercise 2. The allyl radical. 19 Exercise 3. Rumer basis set of VB structures. 23 Exercise 4 : Radical character of Ozone 25 1

2 Starting with XMVB XMVB 3.0 1,2 is the ab initio non-orthogonal Valence Bond program that will be used during the tutorials computer exercises. This section introduces a brief procedure to run an XMVB job. For more details, please refer to XMVB manual at A complete XMVB job includes following steps. 1. Integral preparation If you already have the integrals from previous XMVB jobs, you may skip this step to the next one (B) directly. The integrals for XMVB job can be either obtained from other programs, such as PREINT or GAMESS-US, or formed using the XMVB calculation. The advantage of forming the integrals in advance is that they can be used for many VB calculations as long as the geometry has not changed. In this tutorial we will focus on the use of PREINT and will give only one example of calculation of the integral during the XMVB calculation. For the other options you are referred to the manual. For PREINT, one should prepare a dedicated input file "input.inp" and type the following command to execute it: Ø preint input.inp The integral files "x1e.int" and " x2e.int " along with the output file "input.log" will be obtained as a result. 2. Runing XMVB job After obtaining the integrals, create an input file "file.xmi" for your job. Then run the XMVB job by typing the command: Ø xmvb file.xmi An output file named "file.xmo" will be obtained along with few other files. Keywords in XMVB 3.0 for this tutorial: The input is composed of various sections. The following are the different possibilities. "CTRL", "GEOM", "BFI", "STR", "FRAG", "ORB", "AIM", and "GUS". In this tutorial we will demonstrate the use of "CTRL", "GEO", "STR", "FRAG", "ORB" and "GUS". For the others the user can consult the manual (a) Song, L.; Chen, Z.; Ying, F.; Song, J.; Chen, X.; Su, P.; Mo, Y.; Zhang, Q.; Wu W., XMVB 2.1: An ab initio Non-Orthogonal Valence Bond Program, Xiamen University, Xiamen , China, (b) Song, L.; Mo, Y.; Zhang, Q.; Wu, W. J. Comput. Chem. 2005, 26, 514. (c) Chen Z.; Chen X.; Wu W., J. Chem. Phys. 2013, 138, For details about XMVB, see: 2

3 Each section starts with $NAME_OF_SECTION and ends with $END. Keywords for the $CTRL section: NMUL=n: The multiplicity of the system (2S+1), where S is the total spin of the molecule. The default value is 1, which means a singlet state. ORBTYP=OPTION: Orbital type options used in this tutorial are: HAO: Hybrid atomic orbitals. Orbitals will be described according to either atoms or SAOs (Symmetrized Atomic Orbitals) see FRGTYP for details. In that case $ORB is required and $FRAG is optional. OEO: Overlap enhance orbitals. Orbitals will delocalize in the whole system. In this case $ORB is not required. FRGTYP=OPTION: Fragment type, can be either atom (default) or SAO. If FRGTYP=SAO (Symmetrized Atomic Orbitals) is used, a $FRAG section that consists of definition of subsets of basis functions is necessary. These subsets are defined according to the atom(s) on which the basis functions are centered, and also according to the type of basis functions (s, px, py,... ). GUESS=OPTION: Describes the way to generate or read the initial guess. Options used in this tutorial are: MO: Initial guess of VB orbitals will be obtained directly from molecular orbitals. A $GUS section in the input file is needed. READ: Guess orbitals are read from external file, which should be provided by user under the name: input_file_name.gus. NBO: Guess orbitals are read from external file named "orb.nbo" provided by the user. Earlier preparation is required using the nboprep.exe program, see the manual for details. NAO=n: The last n orbitals in the $ORB section will be taken as active orbitals. NAE=n: Number of active electrons. STR=FULL: Automatically generate a full basis of VB structures according to the given NAO and NAE. If not given, the keyword NSTR as well as $STR section should be given. NSTR=n: Number of VB structures specified in $STR section. ISCF=n: Specifies orbital optimization algorithm. The recommended options for the value n are: 1. Numerical gradients with forward difference approximation are used with the DFP-BFS algorithm. This is an old and slower algorithm, not recommended in most cases. 2. Analytical gradients in terms of basis functions with the L-BFGS algorithm. 3 Recommended for BOVB calculations. 4 3 L. Song, J. Song, Y. Mo, W. Wu, J. Comput. Chem. 2009, 30, Note that this algorithm involves only the first-order density matrix and is not suitable in cases displaying structures that are orthogonal to each other. In this case, it is necessary to use ISCF=1 algorithm for a BOVB calculation. 3

4 5. Analytical gradients in terms of VB orbitals with the L-BFGS algorithm. This is the most efficient algorithm so far for the VBSCF calculations, but cannot be used with BOVB or VBCI calculations. 5 ITMAX=n: specifies the maximum number of iterations. Default number is 200. BOYS: Performs Boys localization. This is essential if VBSCF result is used as the initial guess for VBCI or BOVB calculations. BOYS localization can not be applied to BOVB and VBCI wave functions. BOVB: Performs BOVB calculation. Active and inactive orbitals are different in the different structures while core orbitals can be kept the same in the different structures if NCOR keywords is used. GUESS=READ option is strongly recommended, providing as a guess file a converged set of VBSCF orbitals which were further transformed by using Boys localisation. VBCIS: VBCI with single excitations from active and inactive (but not from core) orbitals. GUESS=READ option is necessary, providing as a guess file a converged set of VBSCF orbitals further transformed using Boys localisation. In addition the NCORE keyword is often used as well. NCOR=n: The first n orbitals in the $ORB section will be taken as core orbitals in VBCI and BOVB calculations. That is for BOVB calculation these orbitals will be kept as VBSCF orbitals and will be the same for all structures. For VBCI or VBPT2 these orbitals will be kept frozen and no excitations will be carried out from them. If the integrals are not given than additional keywords are required in the CRTL section: BASIS=name of basis-set - provides the basis set that will be used for the calculation. INT=CALC: calculates the integrals for the calculation NCHARGE=n If the system is charged NCHARGE=n keyword must be added and n would be the charge. Additional sections used in this tutorial: $GEO section - in which the structure geometry is provided. $STR section - in which the VB structures are defined. $FRAG section - in which the fragments are defined. $ORB section - in which the VB orbitals are defined. $GUS section - in which the guess is defined. 5 Z. Chen, X. Chen, W. Wu, J. Chem. Phys 2013, 138,

5 VB Computer Exercises Exercise 1. The HF molecule. This is a preliminary exercise. All the files have been prepared and commented by the tutors, so the participants will only have to read the (.xmi) input files to understand its format, run and analyze the (.xmo) output files. Files contained in the ex1 folder: - hf-int.inp : input for the preint program. Generate the integral files (x1e.int and x2e.int), and the Hartree-Fock MOs that can be used as guess orbitals. - hf-vbscf.xmi (and hf-vbscf-uncommented.xmi): input for the xmvb program, run a full structure VBSCF calculation. The uncommented file is given for its readability, except for the comment lines the two files are identical. - hf-bovb.xmi: input for the BOVB calculation. It was been created using the hf-vbscf-uncommented.xmi file as a template, and the modifications are commented. - ANSWERS: sub-folder containing all inputs and outputs for this exercise (run by the tutors). 1. Open and read the hf-int.inp file, to see the format (that is simple). Then run the Hartree-Fock calculation using the following command: Ø preint hf-int.inp The preint program generates several files, including the integral files (x1e.int and x2e.int) that are necessary for subsequent XMVB calculations. Open the hf-int.log output file. Note the final Hartree-Fock energy. Inspect the Molecular Orbital (MOs), and in particular determine: a. which MO(s) can be considered as core orbital(s)? b. which MO(s) can be used as guess orbitals for the valence lone pairs of the Fluorine atom in the VB calculation? c. which MO(s) can be used as guess orbital(s) for the active VB orbitals, i.e. the localized orbitals that will described the F H bond? d. which basis-functions are not expected to mix with each other (while contributing to the MO's) in this geometry based on symmetry? The answers to these questions are provided in the $gus and $frag section of the VBSCF input file. The following is a reproduction of the hf-vbscf.xmi input file with comments: HF-LVBSCF # The first line is a comment line, you can write what you want here. $ctrl str=full nao=2 nae=2 # generate all VB structures with 2 active orb. and 2 active elec. # the last two VB orbitals in the $orb section will be the # active orbitals iscf=5 # VBSCF algorithm with reduced density matrix (recommended), not # compatible with BOVB. iprint=2 # Printing option itmax=2000 # Maxinum number of iterations is set to 2000 (recommended). orbtyp=hao frgtyp=sao # Construct VB orbitals with HAOs, fragmented by atom guess=mo # Read the Hartree-Fock MOs (from the hf-int.log file) and use them as # guess orbitals. With this option, a "$gus" block is needed. boys # Request boys localisation at the end of the VBSCF procedure. # This option is strongly recommended when subsequent BOVB or # VBCI calculations are done. $end $frag 1*4 # 4 "fragments" (subsets of basis functions), each containing basis 5

6 # function centered on 1 atom spzdxxdyydzz 2 # First subset of basis functions: contains s, pz and dxx dyy dzz # atomic orbitals, centered on atom number 2 (Fluorine). # The numbering of atoms is the one in the "hf-int.inp" file # (number 2 is the Fluorine) pxdxz 2 # Second subset: contains px type of basis functions centered on atom 2 pydyz 2 # Third subset: contains py type of basis functions centered on atom 2 spz 1 # Fourth subset: contains s and pz type of basis functions centered on # atom 1 (hydrogen) $end $orb 1*6 # 6 orbitals, each developed on 1 subset of basis functions, the # latter being defined in the $frag section. 1 # Orbital 1, defined on subset number 1 from $frag 1 # Orbital 2, defined on subset 1 2 # Orbital 3, defined on subset 2 3 # Orbital 4, defined on subset 3 # The last two orbitals (5 and 6) are the active orbitals 1 # Orbital 5, defined on subset 1 4 # Orbital 6, defined on subset 4 $end $gus # Define which Hartree-Fock MOs (HF-MO) are used as guess orbital for # each of the VB orbitals. In each line, the first number corresponds # the VB orbital (as defined in $orb section), and the second number to # the HF-MO (as defined in the hf-int.log file). 1 1 # HF-MO 1 is used as guess for VB orbital 1 # It ll set as core orbital in the subsequent BOVB calculation 2 2 # HF-MO 2 is used as guess for VB orbital # HF-MO 4 is used as guess for VB orbital # HF-MO 5 is used as guess for VB orbital # HF-MO 3 is used as guess for VB orbital # HF-MO 3 is used as guess for VB orbital 6 # The same HF-MO is used as guess for VB orbitals 5 and 6, because # these orbitals are active in the VB calculation and thus localized # on F or H atoms, while the HF-MO is delocalized on both atoms $end 2. Inspect the VBSCF input file hf-vbscf.xmi (also reproduced in the previous page). 6 Take all the necessary time to carefully read and understand all comments, and each section of this input file. This is necessary to be able to perform the following exercises. In particular, check your answers for point 1. Then run the VBSCF calculation using the following command: Ø xmvb hf-vbscf.xmi Open the output file (hf-vbscf.xmo), and inspect it. Then answer the following questions: a. What is the VBSCF total energy? Compare it with the Hartree-Fock energy (from the hf-int.log file). Comment? b. What are the weights of the VB structures? Draw on paper a schematic representation of the three structures. Are the computed structure weights compatible with your chemical intuition? c. What is the value of the overlap between covalent and ionic structures? d. Inspect the optimized VB orbitals, and in particular check that the active orbitals are localized either on the fluorine or the hydrogen atom Inspect the BOVB input file (hf-bovb.xmi), look in particular at the differences with the VBSCF input file (commented). Create a guess orbital file for the BOVB calculation from the converged VBSCF orbitals using the following command: Ø cp hf-vbscf.orb hf-bovb.gus Then run the BOVB calculation using the following command: Ø xmvb hf-bovb.xmi Open the output file (hf-bovb.xmo), and inspect it. Look in particular to the structure definition ( WEIGHTS OF STRUCTURES section), and comment the difference with the VBSCF calculation. Then answer the following questions: 6 The same file without the comments ( hf-lvbscf-uncommented.xmi ) is also provided for a better readability 7 Note that the optimized orbitals are displayed twice. We recommend to analyze the second set ( ORBITALS IN PRIMITIVE BASIS FUNCTIONS ), because the type of each basis function (s, px, py, ) is also shown. 6

7 a. Compare the value of the VBSCF and BOVB structure weights, and comment. b. Compare the value of the VBSCF and BOVB total energies, and comment. Exercise 2. Ozone and sulfur dioxide In this exercise, the π systems of O 3 and SO 2 will be studied and compared, at the σ-d- VBSCF and σ-d-bovb levels. These acronyms mean that all π-type electrons/orbitals are taken as active (and thus localized), and all σ-type electrons/orbitals are inactive and delocalized over the whole molecule (described through MOs delocalized over the whole molecule). The σ-d-vbscf input files are provided, so you simply have to run this calculation. Then, using the latter files as template, you will have to create the σ-d-bovb input files. The contrasted reactivity of O 3 vs. SO 2 will then be explained from the analysis of the BOVB calculations. The VBSCF calculations are thus in this case only intermediate calculations, to generate converged VBSCF orbitals that will be used as guess orbitals for the following BOVB calculations. 8 Prior to doing this exercise it is recommended to turn to paper exercise Run the preint calculation for Ozone. 9 In the output file (o3-int.log), identify: which MOs can be used as guess for: a. the core orbitals in the VB calculations ; b. the inactive (valence σ) orbitals ; c. the active (π) orbitals. The answers are provided in the $gus section of the o3-vbscf.xmi file. 2. Analyse the σ-d-vbscf input file (o3-vbscf.xmi), and in particular see that the π system only is described at the VB level using orbitals localized on one atom, while the σ system is described as inactive using delocalized MOs. Check your answers for question 1 in the $gus file. 3. There are 3 covalent and 3 ionic structures in a complete basis of nonredundant VB structures (Rumer basis), 10 for the description of the π system of O 3 and at the VB level (4 active electrons in 3 active orbitals). Draw on paper a proposition for these 6 structures. 11 Which structure(s) should be neglected in the BOVB calculation? Run the VBSCF calculations using STR=FULL keyword to check your result. 4. Create the BOVB input file (o3-bovb.xmi) using the VBSCF input files as templates, and the output file for a selection of structures. 12,13 While preparing the file try to think carefully which orbitals should be different for different structures and which 8 It is highly recommended to always use converged VBSCF orbitals as guess for the BOVB calculations, to obtain a physically meaningful wavefunction. 9 preint o3-int.inp 10 See also paper exercise n 4 about Rumer basis. 11 Note that this is the same case (4 electrons in 3 orbitals) as for the SN2 reaction. 12 We will retain in this exercise only the structures that have a weight larger than 2% at the VBSCF level in at least one of the two molecules (O 3 or SO 2). In any case, you should never include in a BOVB calculation structures that have a weight lower than 1%, as this may lead to convergence issue. The selection of structure will be defined in a $str section in the VBSCF input file, and the keyword str=full should change to nstr=x (where X is the number of structures in the BOVB calculation) in the $Ctrl section. 13 Don t forget in particular to change «iscf=5» (algorithm most suitable for VBSCF calculations) to «iscf=2» (algorithm for the BOVB calculations). 7

8 orbitals can be kept the same for the different structures 14 Run the σ-d-bovb calculation for O 3, using the orbitals obtained from the previous VBSCF calculation as a guess. 15 Reactivity: Compare the structure weights for O 3 with the results you obtained in paper exercise 4. Comment. 5. Repeat points 1-4 for the SO 2 molecule. 6. Reactivity: Compare the structure weights for O 3 and SO 2. How could you explain the high reactivity of ozone, vs. the very low reactivity of SO 2? 16 Reference: J. Am. Chem. Soc., 2010, 132, Exercise 3. Benzene and Anthracene In this exercise, we will use a VB description with covalent structures only. Both localized active π orbitals ( HAOs ) and delocalized π orbitals ( OEOs ) will be used in two different VBSCF calculations. 17 The numbers on the pictures below corresponds to the numbering of the carbon atoms in the input files A: BENZENE. 1. The c6h6-5str.xmi file is designed to perform a VBSCF calculation of Benzene with the π system as active, and the σ system as inactive and described by orbitals delocalized over the whole molecule. a. Inspect the $frag and $orb sections, and fill the $str section with a definition for the two Kekulé and three Dewar covalent structures of Benzene. b. Run the preint calculation, then fill the $gus section in the c6h6-5str.xmi file. c. Run the XMVB calculation for the c6h6-5str.xmi file, check the respective weights of the Kekulé and Dewar structures, and comment. 2. A complete description of Benzene includes many more structures. The "c6h6-full.xmi" file is designed to perform a VBSCF calculation of Benzne with π system 14 You will have to add the NCOR=X option in the $Ctrl section, where X is the number of core orbitals in the calculations. These «core» orbital will be reoptimized at the BOVB level, but they will be the same in the different structures. The other ( inactive and active ) orbitals will be different in each structure at the BOVB level. 15 Before running the BOVB calculations, copy the converged VBSCF orbitals to the guess orbital file for the BOVB calculation (cp o3-vbscf.orb o3-bovb.gus). Don t forget to add GUESS=READ in the $CTRL section in the BOVB input file and change iscf=5 to iscf=2. 16 In the references below, it is shown that the reaction barriers of dipolar cycloadditions correlate well with the biradical character in the 1,3 dipole. 17 A VBSCF calculation using a complete basis of covalent structures and delocalized active π orbitals is also called the Spin Coupled VB wave function. 8

9 as active including all the relevant structures. Run the calculation, 18 using the converged VB orbitals of the previous calculation ( c6h6-5str ) as guess. Compare the total energy of the two calculations and comment. 3. Inspect the c6h6-kek-oeo.xmi file. It is designed to perform a Spin- Coupled VB calculation, that is to say a VB calculation using Overlap Enhanced Orbitals (OEOs), 19 with the two Kekulé structures of Benzene as basis of structures. Run the calculation, 20 using the converged VB orbitals of the previous calculation ( c6h6-5str ) as guess. 21 Then: a. Compare the converged VB orbitals for the two calculations, and comment; b. Compare the total energy for the two calculations, and comment. B: ANTHRACENE. 1. There are 4 Kekulé type of structures for Anthracene which are usually used. Draw them on paper. 2. Inspect the c14h10-kek.xmi file. It is designed to perform a VBSCF calculation using the 4 Kekulé structure of Anthracene. From the $Str section, check your answer for point 1. Run the calculation. 3. Create a c14h10-kek-oeo.xmi file using the c14h10-kek.xmi file as template, to perform a Spin-Coupled VB calculation for the 4 Kékulé structures of anthracene. Run the calculation. a. Compare the total energy for the two calculations. b. Compare the structure weights for the two calculations. Comment. Optional Advanced Exercise In the folder DifferentAtomOrder you will find input tile for anthracene with the same geometry but a different order of the atoms as depicted in the figure below. 1. Is there any need to change the VB input file? in which sections? 2. Do the required changes and rerun the VB calculation with the 4 Kekulé structures. 3. What do you think is the advantage of having this order of atoms? 18 You do not need to modify this file. 19 OEOs are simply VB orbitals that span the whole basis of functions. 20 You do not need to modify this file. 21 To get chemically meaningful results in terms of structure weights, it is absolutely necessary to use converged VBSCF orbitals as guess for any calculation involving OEOs. 9

10 4. Use str=cov in the $ctrl section and perform a calculation which includes all the covalent structures. Use also the sort keyword which will sort the VB structures in descending order according to the coefficients. To avoid a long calculation use the NCOR option and use the guess from the previous calculation. Can you find the 4 structures you drew earlier? Are they indeed the most important ones? Can you comment on the energy lowerings by orbital optimization and by involving of more structures? 5. Now perform the same calculation with the original numbering of the atoms. Look at the weights of the different structures. Can you comment? Exercise 4. Identity hydrogen abstraction reaction The chemical reaction studied in this exercise is: H 3 C + H CH 3 [H 3 C--H--CH 3 ] H 3 C H + CH 3 (1) The geometries for the reactant and TS states are provided in the computer files. A template xmi input file for the TS is provided, without the structure block which has to be filled by the participants. Inspect the file and make sure you understand the guess block. 1. Basis set of structures: a. Examine the input files. What are the active orbitals? How many electrons occupy them? b. This is a 3-electron/3-orbital problem. Propose one covalent and two ionic structures for the reactants R (H 3 C + H CH 3 ), draw a picture corresponding to each of them (using the usual convention of drawing of VB structures), and write below their mathematical expression. c. Do the same for the product (P) structures (H 3 C H + CH 3 ) d. Draw the two additional ionic structures that are necessary to get a complete basis set. Hint: you can check your result by performing a VBSCF calculation using the STR=FULL keyword. 2. At the transition State geometry: a. Compute a VBSCF wave function including only the three structures corresponding to the reactants. b. Compute a VBSCF wave function including the full basis of structures. c. Deduce from the total energies of the two previous calculations the resonance energy resulting from the R-P mixing at the transition state. 3. At the reactant geometry: complete the guess block and compute a VBSCF wave function including the full set of VB structures. 4. Using previous results, compute the reaction barrier for the identity hydrogen abstraction reaction at the VBSCF level. Compare with the Coupled-Cluster CCSD value of 22.9 kcal/mol and CCSD(T) value of 21.6 kcal/mol in the same basis set. 5. Repeat points 2-4 at the VBCIS level Don t forget to select «iscf=1» algorithm with the VBCIS keyword. 10

11 Exercise 5. The H 3 Si-H vs. H 3 Si-Cl Bonds Sometimes the MO guess is not good enough or not easy to properly assign. In this exercise, we will demonstrate the use the NBO orbitals as guess. This requires preparation of an NBO type of calculation. The NBO calculations were prepared for you in advance and the corresponding files can be found in the designated NBO folders. Start with the H 3 Si-H bond. 1. Convert the NBO orbitals into a format readable for the VB program using the following command: Ø nboprep.exe sih3h-nbo.log NBO The nboprep.exe program generates a file named "orb.nbo" that contains the NBO orbitals. You need to have this file available in the directory where you run the VB calculation. 2. Open the sih3h-nbo.log file, and inspect the NBO orbitals in the Table entitled: "Natural Bond Orbitals (Summary)". In particular determine: a. which NBO(s) describe the occupancy of core electrons(s)? b. which NBO(s) can be used as guess orbitals for the inactive Si-H bonds in the VB calculation? c. which NBO(s) can be used as guess orbital(s) for the active VB orbitals? 3. Inspect the VBSCF input file "sih3h-vbscf.xmi". Make sure you fully understand the $gus section and that there is agreement between this section and your answers to point Perform the following calculations: a. Provide the 3 VB structures that describe this system (one covalent and two ionic) and compute the VBSCF wave function. b. Compute a VBSCF wave function including only the covalent structure. c. Deduce from the total energies of the two previous calculations the resonance energy (RE) resulting from the covalent-ionic mixing. Note that the resonance energy of a bond is defined as the difference between the energy of the ground state and the energy of its most stable VB structure. d. Calculate the energy of the respective radicals at dissociation using an ROHF calculation. For your convenience the input for the SiH3 radical has been prepared for you in the Radicals directory. e. Calculate the dissociation energy of the "exact" ground state (De) and the dissociation energy of the covalent state (Dcov). Based on these dissociation energies and the RE obtained in point c what can you say on the character of this bond? 5. Repeat the calculation of 4a using the BOVB method and answer points 4a-4d at the BOVB level. 6. Repeat points 1-4 for the H3Si-Cl bond 7. Compare the two bonds. Reference: Chem. Eur. J. 2005, 11,

12 Exercise 6. Cl-CH 3 -Cl vs. Cl-SiH 3 -Cl This exercice is optional; you might experience convergence issues for the wavefunction Cl-CH 3 -Cl is the pentacoordinated transition state in the identity S N 2 reaction while the Cl- SiH 3 -Cl is a stable species. In this exercise we try to partially understand the difference between these two systems while using NBO orbitals as a guess. 1. Start with the Cl-SiH3-Cl system. Inspect the input file for the VBSCF calculation and in particular the $gus section. Make sure you understand all the choices especially the active orbitals. Ø What was the problem when choosing a guess for the active orbitals? How was it solved? A different solution can be found in "Optional Advanced Exercise". 2. Complete the $str section by providing the 3 most important VB structures. 3. Perform the calculation. Don t forget to provide the orb.nbo guess file which should be created. 4. Analyze the results: a. Calculate the difference between the different structures. b. look at the weights of the different structures. c. based on the results you obtained, draw a schematic diagram that describes the curve crossing situation in this case. 5. Now move to the Cl-CH3-Cl system. In the NBO directory you will find two different NBO calculations. One termed "ClCH3Cl-nbo_TSGeom.com" which contains the real TS geometry and the other termed "ClCH3Cl-nbo_forGuess.com" where the C-Cl distance was made shorter. Inspect the input and output files. What is the difference? We will use the NBO's of the second "ClCH3Cl-nbo_forGuess.com" file. Do you understand the reason? 6. Repeat points 1-3. Can you explain difference in the behavior of the two systems? Optional Advanced Exercise 7. Inspect the active orbitals, especially the central one on carbon/silicon, explain that why the s basis functions are missing there. 8. Considering that how many ways we may have to improve the ground energy by involving the s basis functions. If an extra s-type orbital is introduced, how many structures should we add? Prepare the input file for the 5 structure calculation. Perform calculations, look at the weights of the additional structures in the two systems, and explain the difference. 9. Recollect the BOVB idea, and considering that should the central orbital keep the same when make a bond with the left and right Cl atoms? Split the central orbital, prepare the input file, compare the total energy and wave function with that obtained in step 7, and make a comment. Exercise 7. Allyl Radical In this exercise we will look at the wavefunction in more detail including the determinants that compose it and better understand the choice of VB structures. We will also calculate both the 12

13 ground and the excited states of allyl radical and understand the difference in their properties. Prior to the calculations do paper exercise The covalent state of allyl radical a. Complete the input file "Allyl_cov.xmi" and calculate the VBSCF wave function for the covalent state of the allyl radical using the covalent structures you have chosen in paper Ex What are the weights of the different VB structures. Was that expected? b. What is the wavefunction in determinant description? What is the spin population? Do the calculated results agree with your answer for paper Ex. 2.2? c. Repeat the calculation for the first excited state using "nstate=1". What is the wavefunction? Use determinant description. What is the spin population in this case? Was that expected? 2. Allyl radical adiabatic energy and weights: a. Calculate a complete VBSCF wave function for allyl radical (the adiabatic wavefunction) using a complete set of structures covalent and ionic (Paper Ex. 2.1 and 2.3). Compare the weight and energies of the different ionic structures. can you understand the difference? Do the results agree with your selection of structures in paper Ex. 2.3? b. calculate the BOVB wave function for allyl radical (remember to use the guess orbitals obtained at the VBSCF level). Compare the weights obtained at the VBSCF and BOVB levels. 3. Calculation of resonance energies: In order to calculate the resonance energy you should calculate the wave-function that corresponds to only one of the Lewis (diabatic) structures of allyl radical. To do so you should include in the wave-function only one covalent structure, and the ionic structures associated with this covalent bond. An input file for the Lewis structure has been prepared for you "Allyl_WrongLewis.xmi". The input however consist of a mistake in one of the sections. Allocate the mistake and calculate the wave function at the VBSCF and then BOVB levels. Deduce what is the resonance energy of the allyl radical at the BOVB level. Remember - the resonance energy is calculated as the difference between the adiabatic state and the Lewis (diabatic) state of the allyl radical. Exercise 8. CycloButadiene In this exercise an input file that aims to calculate the covalent state of cyclobutadiene was prepared for you. The file however contains a mistake. Find what is the mistake and give two different solutions. For your convenience, the energy of this system should be au. Exercise 9. Resonance energy of formamide 13

14 In this exercise, you ll have to prepare yourself all your input files and do all the necessary calculations from scratch. Compute the resonance energy of formamide at the σ-d-bovb level of theory (use the 6-31G* basis set). Compare it with the rotation barrier of formamide (~16 kcal.mol 1 ). The only data given you in this exercise are the coordinates which are available below as well as in the file formamide.xyz Formamide geometry: C O N H H H Exercise 10. Diels-Alder reaction of 1, 3-butadiene and ethylene In this exercise, we will calculate reaction barrier (E ) and reaction enthalpy (ΔH) of the reaction of 1, 3-butadiene and ethylene by means of VBSCF method. We will learn how to select VB structures, and how to divide the molecules into fragment such that the VB structure can be interpreted in the framework of classical VB theory, at the same time keep the total wave function as compact as possible. This example also exemplified some tips in the generation of a suitable initial guess. 1. $FRAG section At the reactant geometry, it is apparent that we can divide the molecule into two fragments, namely that 1, 3-butadiene and ethylene. Thus we can use this fragments for the whole reaction path. 2. $ORB section Localized inactive and active bond orbitals might be obtained if we use localized orbitals as initial guess. NBO is also recommend for this example. Inspect how many active orbitals and electrons should be involved for this reaction? Think about that how to generate meaningful hybrid atomic orbitals from NBO results. 3. $STR section For the purpose of obtaining a smooth potential energy surface, it is apparent that we should use an unchanged set of VB structures when the geometry changed. Draw down the most important structures at the reactant and the product geometries. Draw down the remaining two Dewar structures, which may contribute at the transition state geometry. According to the definition of fragment, what ionic structures should be selected also to improve the wave function? 4. Prepare an input file with 5 covalent structures. Inspect the atom ordering; divide the molecule into six fragments to read NBOs as initial guess for both inactive and active VB orbitals. 5. Prepare an input file with 5 covalent structures. Divide the molecule into two fragments, read the orbitals from step 4 as initial guess. 6. Prepare an input file with 5 covalent structures and important ioinic structures. Divide the molecule into two fragments, read the orbitals from step 5 as initial guess. 7. Prepare an input file with full VB structures, read the orbitals from step 6 as initial guess. Use sort option, and compare the weights by select structure and full structure calculations. What is the sum of the weights of selected VB structures in the full structure results. 8. Plot a figure shows the variety of weights of the reactant, product and intermediate state functions along the reaction path. Here, the reactant state function is defined the summation of the VB structure at the reactant geometry. 14

15 VB Paper Exercises Exercise 1. The lone pairs of H 2 O This exercise aims at comparing two descriptions of the lone pair of H 2 O: (i) the MO description in terms of non-equivalent canonical MOs and (ii) the rabbit-ear VB description in terms of two equivalent hybrid orbitals. 23 Figure 1: Lone pairs of H2O represented in MO and VB theory 1. Focusing on the lone pairs only, write the four-electron single- determinants Ψ MO and Ψ VB. 2. Expand Ψ VB into elementary determinants containing only n and p orbitals, eliminate determinants having two identical spin-orbitals, and show the equivalence between Ψ MO and Ψ VB. 3. We now remove one electron from H 2 O. Write the two possible VB structures Φ 1 and Φ 2 in the VB framework. By convention, one may write the doubly occupied lone pair first, then the singly occupied one. 4. The two ionized states are the symmetry-adapted combinations and. Knowing that the sign of the Hamiltonian matrix element is negative, give the energy ordering of the two ionized states. 5. By expanding the two ionized states into elementary determi- nants (dropping the normalization constants), show that they are equivalent, respectively, to the MO configurations nn p and p p n. Exercise 2. Allyl radical In this exercise, the participants will try to write the VB structures of allyl radical, understand the spin density polarization of allyl radical with VB theory, and try to select a subset of VB structures from the complete set. Figure 2: Allyl radical (showing also the three p active orbitals) 23 For further reading, see S. Shaik and P.C. Hiberty, The Chemist s Guide to VB theory, Wiley, Hoboken, New Jersey, 2008, pp

16 1. Covalent structures of the allyl radical: a. What are the three possible covalent structures for the allyl radical molecule? b. Show that the third structure can be expressed as a linear combination of the first two structures, and thus that only two of the three covalent structures form a complete basis of non-redundant structures (Rumer basis). 2. Understanding the pattern of spin density distributions in the allyl radical as found in EPR spectroscopy. a. Express the wave functions of the non-redundant structures (from 1a) of the allyl radical in terms of the VB determinants. Write the wave function of the ground state of the allyl radical as a negative combination of the wave functions Φ L and Φ R of the nonredundant structures. Based on the expression of the spin density ρ k S on atom k : propose the spin density distribution in the allyl radical. In the above equation N is a normalization constant, c is the coefficient of the VB determinant in the wave function and δ is either +1 or 1 depending on whether the electron which is located on atom k in the i th determinant is α or β spin, respectively. b. Show pictorially : i. Why the spin density is polarized? ii. What would be the spin density in the excited state of the allyl radical (taking into account that resulting wave function is a positive combination of the Φ L and Φ R wave functions)? iii. What would be the spin density pattern in pentadienyl radical? What are the possible ionic structures for the allyl radical? Based on your chemical knowledge, propose a selection subset of the most chemical meaningful covalent + ionic structures. Exercise 3. Rumer basis set of VB structures Rumer s method is an easy way to generate a complete and non redundant set of VB structures. The method is the following : 1. For covalent structures : put the orbitals on an imaginary circle, even if the molecule is not a ring, with one electron per orbital. Then generate all possible covalent structures not displaying crossing bonds. 2. For ionic structures : choose a distribution of charges in the imaginary circle (i.e. one orbital bears two electrons end another one is empty), then apply Rumer s rule on the rest of the system. Examples : benzene 1,3-butadiene cyclo-octatetraene 24 Hint: reason on the total spin-alterned determinants as being the major one. 16

17 Questions for this exercise : 1. Draw the Rumer basis set of covalent VB structures for benzene, including «long-bond» structures. 2. For For 1,3-butadiene, draw the Rumer basis set of covalent VB structures, then the full basis set of covalent + ionic VB structures. 3. Draw the Rumer basis set of covalent VB structures for cyclooctatetraene. Exercise 4 : Radical character of Ozone 1. Propose a complete basis of non-redundant VB structures of the π electronic system of ozone and write their wave-functions. 2. Express the π-mo wave function of ozone in terms of the VB structures of the preceding exercise, given the list of the π-mos is: A. At the Hartree-Fock approximation: B. At the Hückel approximation: Hint: Write a single-determinant MO wave function based on these orbitals. Develop it into the basis of atomic orbitals, to get an expression in terms of VB structures. It is useful to use the half determinants methods. 3. Compute by hand the weights of the different structures at the Huckel approximation (neglecting all overlaps). What is the radical character of ozone according to simple MO theory? 4. The wave function of ozone is given by: Express this wave function in terms of VB structures, and show that CI has the effect of increasing the weight of the diradical structure, and lowering those of the ionic structures, especially the 1,3-dipolar ones. 17

18 ANSWERS to the VB Paper Exercises Exercise 1. The lone pairs of H 2 O 18

19 Exercise 2. The allyl radical. 1. a. b. 19

20 2. a. b. 20

21 21

22 3. 22

23 Exercise 3. Rumer basis set of VB structures

24

25 Exercise 4 : Radical character of Ozone 1. The complete set of the non-redundant VB structures of the π electronic system of ozone are described below (Table 1). Grey lines represent σ electrons. The overlaps between AO-determinants have been neglected in the calculations of the normalization factors in Φ 1 -Φ 3. Table 1. ionic ionic diradical di-ionic dipolarionic dipolarionic 2. A single-determinant MO wavefunction of ozone would look as follows: We will demonstrate here two different ways of solving the problem: A. At the Hartree-Fock approximation, using the Half determinant Approach: A simple way to solve it is by using the half determinant approach. Thus, the VB functions - involve three different half-determinants, of or spin: ; ;. Knowing the expressions of π 1 and π 2 we can find the coefficients of the three halfdeterminants by using the equation: where r is a given set of AOs P is a permutation between the indices µ and ν of the AOs in the half determinant. For example, the coefficient of ; in ; is given by the following equation:

26 The coefficients of these half determinants are listed in the following Table: coefficient From these values, the coefficients of the full AO-determinants are calculated. For example, the determinant is made of the half-determinants and. Its coefficient is therefore: (-0.542) (-0.523). Note that the order of the orbitals in a half-determinant is important: two halfdeterminants that are related to each other by a permutation of a pair of orbitals have opposite coefficients. Following these rules, we can find the coefficients of all the different full AO-Determinants in the MO wave-function. Determinant Coefficient These coefficients, represent expansion of the the Hartree-Fock wave function in terms of the AO-based determinants. Now looking at the expansion of the VB structures (Table 1) it is easy to express the Hartree-Fock wave function in terms of VB structures as follows: B. At the Hückel approximation using the electron-hole approach: A different way to solve it is by using the electron-hole approach. We will use it while solving with the Hückel wave function to avoid redundancy of solution. Thus, the list of the π-mos at the Huckel approximation is: 26

27 The Hückel wavefunction of Ozone is therefore: We can simplify the problem by using the electron/hole equivalence. Thus, we simply replace holes by electrons and vise versa in getting a function of, which we will then expande in terms of VB structures. Finally, we will perform the back holeelectron transformation in the VB representation to get the final result. This way we transform a 4e-3c problem into 2h-3c one. Expanding into AO determinants, we get: Doing the electron-hole back transformation, we get: 3. The weights of the VB structures while neglecting overlap (at the Huckel approximation) are simply the square of the coefficient leading to. Thus, according to simple MO theory the radical character of ozone is only(!) 12.5%. 4. The diexcited configuration is expanded in the same way as the Hartree-Fock configuration (previous point A). The coefficients of the half-determinants are: coefficient Combining the expression for the ground state and the diexcited states leads to the following expantion of the CI wavefunction: 27